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User blog:Cheetahrock63/Hypercomplex Blog: Schl
Home Simple-ish way to find out if a tiling is hyperspherical, Euclidean, or hyperbolic. The Schläflian of a polytope is the determinant of the Schläfli matrix, which can be found from the Schläfli symbol. As mentioned in my last post, the sign of the Schläflian of a tiling can be used to find out what type of tiling it is. This rule is known as Schläfli’s Criterion. A tiling having a positive Schläflian implies that it is a hyperspherical tiling (i.e. a polytope), a tiling with a Schläflian equal to 0 implies that it is Euclidean, and a tiling with a negative Schläflian implies that it is hyperbolic. Hyperbolic tilings can be further divided into compact, paracompact, and noncompact varieties. All Schläfli matrices that are proper submatrices of a compact hyperbolic tiling’s Schläfli matrix have a positive Schläflian. All Schläfli matrices that are proper submatrices of a paracompact hyperbolic tiling’s Schläfli matrix have a nonnegative Schläflian. And all Schläfli matrices that are proper submatrices of a noncompact hyperbolic tiling’s Schläfli matrix have a real Schläflian. (This does mean that all compact hyperbolic tilings are paracompact, all paracompact hyperbolic tilings are noncompact, and, confusingly, all compact hyperbolic tilings are noncompact. One way to reconcile this in your mind is to think of the prefix “'non-'” as short for “'no't n'ecessarily”—exactly like in the case of nonassociative algebras.) I call cases of tilings that are not hyperspherical, Euclidean or hyperbolic such as the complexgon \{3+2i\} or metalgons like goldengons, “'synkrobolic tilings” or “'synkrotopes'”. Generally synkrotopes have a nonreal complex Schläflian, but there are exceptional cases where synkrotopes can have real Schläflians. Complex numbers s such that (s^2 \notin {\mathbb {R}}) \land (\cos (\frac{\pi}{s}) \in {\mathbb {R}}) are examples of what I call "exceptional synkronumbers" since the Schläflian of a polygon with Schläfli symbol \{s\} is real. The set of all exceptional synkronumbers \mathfrak{S} = \{z \in {\mathbb {C}}|(z^2 \notin {\mathbb {R}}) \land (\sin^2 (\frac{\pi}{z}) \in {\mathbb {R}})\} \cup \{\frac{1}{n}|n \in \mathbb {Z} \backslash \{0\}\} include all of such s , as well as all reciprocals of a (nonzero) integer. Polygons whose Schläfli symbols are \{\frac{1}{n}\} where n is a nonzero integer are known as monogons. Similar to how complexgons whose symbols are \{s\} where s is an exceptional synkronumber have a real Schläflian yet are synkrobolic, monogons have a Schläflian of zero but they are most often realized as hyperspherical tilings. All exceptions to Schläfli’s Criterion are tilings that have at least one entry that is an exceptional synkronumber. Schläfli symbol of a regular polytope (entries are elements of (\mathbb {C} \backslash (\{0\} \cup \mathfrak{S})) \cup \{\infty\} ) to Schläfli matrix: \{p, q, r, ...\} \to \begin{bmatrix} 2 & -2 \cos (\frac{\pi}{p}) & 0 & 0 & ...\\ -2 \cos (\frac{\pi}{p}) & 2 & - 2\cos (\frac{\pi}{q}) & 0 & ...\\ 0 & - 2 \cos (\frac{\pi}{q}) & 2 & - 2 \cos (\frac{\pi}{r}) & ...\\ 0 & 0 & - 2 \cos (\frac{\pi}{r}) & 2 & ...\\ ... & ... & ... & ... & ...\\ \end{bmatrix} *The Schläfli matrix of a polytope of dimension k is a k\times k matrix. The Schläfli matrix of a point is the empty matrix. The null polytope has no Schläfli matrix. *All diagonal entries of a Schläfli matrix are 2 * a_{k,k+1} = a_{k+1,k} = - 2 \cos (\frac{\pi}{e_k}) where e_k is the k th entry of the Schläfli symbol of a regular polytope. *All other entries are 0 . All uniform polytopes and tilings generated by Coxeter groups have a Schläfli matrixhttps://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram#Schl%C3%A4fli_matrix. However, I currently don’t know how to derive the Schläfli matrices of nonregular uniform tilings such as the 6-dimensional, 7-dimensional, and 8-dimensional Gosset polytopes. Similar to how real uniform polytopes have Schläfli matrices corresponding to their Coxeter groups, complex polytopes have generalized Cartan matrices that correspond to their Shephard groupshttps://en.wikipedia.org/wiki/Complex_reflection_group#Cartan_matrices. In general, the generalized Cartan matrix of a complex polytope (not to be confused with complextopes/synkrotopes) is asymmetric (and is only symmetric if the polytope is a real polytope) and allows for diagonal entries to be not two. The matrix of a polycomtelon with Schläfli symbol {}_p \{\} is i}{p}) . The matrix of the Möbius–Kantor polygon (aka the berylocomgon) is \begin{bmatrix} 1+\frac{1-i\sqrt{3}}{2} & 1 \\ \frac{1-i\sqrt{3}}{2} & 1+\frac{1-i\sqrt{3}}{2}\\ \end{bmatrix} . A list of matrices of complex polytopes are in this post. (I also don’t know how to derive such matrices.) In general, the Schläflian of a simplex of dimension k is k+1 , the Schläflian of a staurotope (hypercube or cross polytope) of dimension k (where k is a natural greater than 0) is 2, and the Schläflian of a rhodotope (cosmotope or hydrotope) with dimension k (where k is a natural greater than 0) is (1-\phi) k + \frac{3+\sqrt{5}}{2} . From there, one can easily observe that there are no polytopes that are analogues of a dodecahedron or icosahedron with dimensionality greater than 4. The difacetal angle of a given polytope k , a generalization of the internal angle of a polygon and the dihedral angle of a polyhedron, can be found from figuring out which tiling of k has a Schläflian of 0 (that is, it is a Euclidean tiling. In general, this tiling is failed). The angle in radians is \tau divided by order of the Euclidean k tiling. For example, the internal angle of a triangle is \frac{\tau}{6}=\frac{\pi}{3} as a Euclidean tiling made up of triangles is an order-6 triangular tiling. The dihedral angle of a tetrahedron is \frac{\tau}{\frac{\tau}{\arccos (\frac{1}{3})}}=\arccos (\frac{1}{3}) since the (failed) Euclidean tiling composed of tetrahedra is an order- \frac{\tau}{\arccos (\frac{1}{3})} tetrahedral tiling. Rank -1: Nullitope Rank 0: Polya Rank 1: Polytela Rank 2: Polygons Complex_schläflian(z).png|Contour plot of Schläflian of \{z\} . Rank 3: Polyhedra Digonal tilings Complex_schläflian(2,z).png|Contour plot of Schläflian of \{2, z\} . Great heptagrammic tilings Complex_schläflian(7/3,z).png|Contour plot of Schläflian of \{\frac{7}{3}, z\} . Pentagrammic tilings Complex_schläflian(2.5,z).png|Contour plot of Schläflian of \{\frac{5}{2}, z\} . Octagrammic tilings Complex_schläflian(8/3,z).png|Contour plot of Schläflian of \{\frac{8}{3}, z\} . Triangular tilings Complex_schläflian(3,z).png|Contour plot of Schläflian of \{3, z\} . Heptagrammic tilings Complex_schläflian(7/2,z).png|Contour plot of Schläflian of \{\frac{7}{2}, z\} . Square tilings Complex_schläflian(4,z).png|Contour plot of Schläflian of \{4, z\} . Pentagonal tilings Complex_schläflian(5,z).png|Contour plot of Schläflian of \{5, z\} . Hexagonal tilings Complex_schläflian(6,z).png|Contour plot of Schläflian of \{6, z\} . Heptagonal tilings Complex_schläflian(7,z).png|Contour plot of Schläflian of \{7, z\} . Octagonal tilings Complex_schläflian(8,z).png|Contour plot of Schläflian of \{8, z\} . Apeirogonal tilings Complex_schläflian(∞,z).png|Contour plot of Schläflian of \{\infty, z\} . Pseudogonal tilings Complex_schläflian(12i,z).png|Contour plot of Schläflian of \{12i, z\} . Complex_schläflian(9i,z).png|Contour plot of Schläflian of \{9i, z\} . Complex_schläflian(3i,z).png|Contour plot of Schläflian of \{3i, z\} . Complex_schläflian(i,z).png|Contour plot of Schläflian of \{i, z\} . Goldengonal tilings Rank 4: Polychora Great stellated dodecahedral tilings Complex_schläflian(5/2,3,z).png|Contour plot of Schläflian of \{3,3,z\} . Small stellated dodecahedral tilings Complex_schläflian(5/2,5,z).png|Contour plot of Schläflian of \{3,3,z\} . Great icosahedral tilings Complex_schläflian(3,5/2,z).png|Contour plot of Schläflian of \{3,3,z\} . Tetrahedral tilings Complex_schläflian(3,3,z).png|Contour plot of Schläflian of \{3,3,z\} . Octahedral tilings Complex_schläflian(3,4,z).png|Contour plot of Schläflian of \{3,4,z\} . Icosahedral tilings Complex_schläflian(3,5,z).png|Contour plot of Schläflian of \{3,5,z\} . Cubic tilings Complex_schläflian(4,3,z).png|Contour plot of Schläflian of \{4,3,z\} . Great dodecahedral tiling Complex_schläflian(5,5/2,z).png|Contour plot of Schläflian of \{5,5/2,z\} . Dodecahedral tilings Complex_schläflian(5,3,z).png|Contour plot of Schläflian of \{5,3,z\} . Triangular tiling tilings Complex_schläflian(3,6,z).png|Contour plot of Schläflian of \{3,6,z\} . Square tiling tilings Complex_schläflian(4,4,z).png|Contour plot of Schläflian of \{4,4,z\} . Hexagonal tiling tilings Complex_schläflian(6,3,z).png|Contour plot of Schläflian of \{6,3,z\} . Order-7 triangular tiling tilings Complex_schläflian(3,7,z).png|Contour plot of Schläflian of \{3,7,z\} . Order-5 square tiling tilings Complex_schläflian(4,5,z).png|Contour plot of Schläflian of \{4,5,z\} . Order-4 pentagonal tiling tilings Complex_schläflian(5,4,z).png|Contour plot of Schläflian of \{5,4,z\} . Order-4 hexagonal tiling tilings Complex_schläflian(6,4,z).png|Contour plot of Schläflian of \{6,4,z\} . Order-3 heptagonal tiling tilings Complex_schläflian(7,3,z).png|Contour plot of Schläflian of \{7,3,z\} . Order-3 octagonal tiling tilings Complex_schläflian(8,3,z).png|Contour plot of Schläflian of \{3,3,z\} . Rank 5: Polytera Small stellated hecatonicosachoral tilings Grand hexacosichoral tilings Pentachoral tilings Hexadecachoral tilings Hexacosichoral tilings Icositetrachoral tilings Faceted hexacosichoral tilings Tesseractic tilings Great hecatonicosachoral tilings Hecatonicosachoral tilings Cubic honeycomb tilings Rank 6: Polypeta Hexateral tilings Penteractic tilings Triacontaditeral tilings Order-4 icositetrachoral tetracomb tilings Tesseractic tetracomb tilings =References= Category:Blog posts